Cyclic Group Actions on Polynomial Rings
نویسندگان
چکیده
We consider a cyclic group of order p acting on a module incharacteristic p and show how to reduce the calculation of the symmetric algebra to that of the exterior algebra. Consider a cyclic group of order p acting on a polynomial ring S = k[x1, . . . , xr], where k is a field of characteristic p; this is equivalent to the symmetric algebra S∗(V ) on the module V generated by x1, . . . , xr. We would like to know the decomposition of S into indecomposables. This was calculated by Almkvist and Fossum in [1] in the casen = 1; see also [6]. They reduced the problem to the calculation of the exterior powers of V , and then gave a formula for these. In this note we accomplish the first part for general n, that is to say the reduction of the calculation of the symmetric algebra to that of the exterior algebra. Many of the results extend to a group with normal cyclic Sylow p-subgroup, in particular to any finite cyclic group. We wish to thank Dikran Karagueuzian for providing the computer calculations using Magma [4] that motivated this work.
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